Array-based genetic analyses start with a large library of cDNAs or oligonucleotides (probes), immobilized on a substrate. The probes are hybridized with a single labeled sequence, or a labeled complex mixture derived from a tissue or cell line messenger RNA (target). As used herein, the term “probe” will therefore be understood to refer to material tethered to the array, and the term “target” will refer to material that is applied to the probes on the array, so that hybridization may occur.
The term “element” will refer to a spot on an array. Array elements reflect probe/target interactions. The term “background” will refer to area on the substrate outside of the elements.
The term “replicates” will refer to two or more measured values of the same probe/target interaction. Replicates may be independent (the measured values are independent) or dependent (the measured values are related, statistically correlated, or reaction paired). Replicates may be within arrays, across arrays, within experiments, across experiments, or any combination thereof.
Measured values of probe/target interactions are a function of their true values and of measurement error. The term “outlier” will refer to an extreme value in a distribution of values. Outlier data often result from uncorrectable measurement errors and are typically deleted from further statistical analysis.
There are two kinds of error, random and systematic, which affect the extent to which observed (measured) values deviate from their true values.
Random errors produce fluctuations in observed values of the same process or attribute. The extent and the distributional form of random errors can be detected by repeated measurements of the same process or attribute. Low random error corresponds to high precision.
Systematic errors produce shifts (offsets) in measured values. Measured values with systematic errors are said to be “biased”. Systematic errors cannot be detected by repeated measurements of the same process or attribute because the bias affects the repeated measurements equally. Low systematic error corresponds to high accuracy. The terms “systematic error”, “bias”, and “offset” will be used interchangeably in the present document.
An invention for estimating random error present in replicate genomic samples composed of small numbers of data points has been described by Ramm and Nadon in “Process for Evaluating Chemical and Biological Assays”, International Application No. PCT/IB99/00734. In a preferred embodiment, the process described therein assumed that, prior to conducting statistical tests, systematic error in the measurements had been removed and that outliers had been deleted.
In accordance with one aspect, the present invention is a process that estimates and removes systematic error from measured values. In another aspect, it is a process for optimizing outlier detection and deletion. A second aspect is a process for detecting and deleting outliers. A third aspect is a process for optimizing outlier detection and deletion automatically. A fourth aspect is a process for estimating the extent of random error present in replicate genomic samples composed of small numbers of data points.
There are two types of systematic error potentially present in hybridization arrays.
Array elements may be offset within arrays. Typically, this offset is additive. It can derive from various sources, including distortions in the nylon membrane substrate (Duggan, Bittner, Chen, Meltzer, & Trent “Expression profiling using cDNA microarrays”, Nature Genetics, 21, 10-14 (1999).
If present, the offset is corrected by a procedure called “background correction”, which involves subtracting from the array element the intensity of a background area outside of the element.
Areas used for calculation of background can be close to the array element (e.g., a circle lying around the element), or distant (a rectangle lying around the entire array). Because offset within an array tends to be specific to individual array elements (even with relatively uniform background), areas close to the element are generally preferred for background correction.
Alternatively, background estimates can be obtained from “blank” elements (i.e., elements without probe material). In this procedure, “background” is defined differently from the more typical method described in the previous paragraph. Theoretically, blank element intensities are affected by the same error factors that affect non-element background areas (e.g., washing procedures) and also by error factors which affect element quantification but which are extraneous to the biological signal of interest (e.g., dispensing errors).
The present invention does not address the issue of background correction. In a preferred embodiment, background correction, as necessary, has been applied prior to estimation of systematic error and outlier detection. In a non-preferred embodiment, the process may still be applied to arrays which have not been corrected for background offset.
In one aspect, the present invention is a process for estimating and removing systematic error across arrays. Contrary to background offset, offset across arrays tends to be proportional.
Offset across arrays can derive from various sources. For microarray studies which use fluorescent labelling, offset factors include target quantity, extent of target labelling, fluor excitation and emission efficiencies, and detector efficiency. These factors may influence all elements equally or may in part be specific to element subsets of the array. For example, quantity of target material may be offset differently for different robotic arrayer spotting pin locations (see Bowtell “Options available—from start to finish—for obtaining expression data by microarray” Nature Genetics, 21, 25-32, p. 31 (1999).
For radio-labelled macro array studies, proportional offset factors include target quantity and target accessibility (Perret, Ferran, Marinx, Liauzun, et al. in “Improved differential screening approach to analyse transcriptional variations in organized cDNA libraries” Gene, 208, 103-115 (1998)).
Time of day that arrays are processed (Lander “Array of hope” Nature Genetics, 21, 3-4 (1999)) and variations in chemical wash procedures across experiments (Shalon, Smith, & Brown “A DNA microarray system for analyzing complex DNA samples using two-color fluorescent probe hybridization” Genome Research, 6, 639-645 (1996)) have also been cited as offset factors.
Prior art methods for removing systematic error are called “normalization” procedures. These procedures involve dividing array element values by a reference value. This reference can be based on all probes or on a subset (e.g., “housekeeping genes” whose theoretical expression levels do not change across conditions). However obtained, the reference can be estimated by one of various summary values (e.g., mean or a specified percentile).
Once systematic error has been removed, any remaining measurement error is, in theory, random. Random error reflects the expected statistical variation in a measured value. A measured value may consist, for example, of a single value, a summary of values (mean, median), a difference between single or summary values, or a difference between differences. In order for two values to be considered significantly different from each other, their difference must exceed a threshold defined jointly by the measurement error associated with the difference and by a specified probability of concluding erroneously that the two values differ (Type I error rate). Statistical tests are conducted to determine if values differ significantly from each other.
All of prior art normalization procedures, however, estimate systematic error outside of the context of a statistical model. Because these informal procedures make implicit (and often incorrect) assumptions about the structure of the data (e.g., form and extent of both systematic and random error), they often fail to adequately eliminate measurement bias and can introduce additional bias due to the normalization procedure itself. In a different scientific context, Freedman and Navidi, in “Regression models for adjusting the 1980 census”, Statistical Science, 1, 3-11 (1986) described the problems inherent in failing to correctly model data that contain measurement error (“uncertainty” in their terminology):                Models are often used to decide issues in situations marked by uncertainty. However, statistical inferences from data depend on assumptions about the processes which generated those data. If the assumptions do not hold, the inferences may not be reliable either. This limitation is often ignored by applied workers who fail to identify crucial assumptions or subject them to any kind of empirical testing. In such circumstances, using statistical procedures may only compound the uncertainty (p. 3).        
In addition to correct removal of systematic error, many statistical tests require the assumption that residuals be normally distributed. Residuals reflect the difference between values' estimated true scores and their observed (measured) scores. If a residual score is extreme (relative to other scores in the distribution), it is called an outlier. An outlier is typically removed from further statistical analysis because it generally indicates that the measured value contains excessive measurement error that cannot be corrected. In order to achieve normally distributed residuals, data transformation is often necessary (e.g., log transform).
In one aspect, the present invention is a process for detecting and removing outliers by examining the distribution of residuals. In another aspect, it is a process for detecting and removing outliers automatically through an iterative process which examines characteristics of the distribution of residuals (e.g., skewness, kurtosis).
As with correction for offset across arrays (normalization), prior art for outlier detection relies on informal and arbitrary procedures outside of the context of a statistical model. For example, Perret, Ferrán, Marinx, Liauzun, et al. “Improved differential screening approach to analyse transcriptional variations in organized CDNA libraries” Gene, 208, 103-115 (1998), compared the intensity of sets of two replicate array elements after normalization. Any replicate set that showed a greater than 2-fold difference (or equivalently, less than a 0.5-fold difference) was regarded as an outlier.
In accordance with one aspect, the present invention is a process for estimating the extent of random error present in replicate genomic samples composed of small numbers of data points and for conducting a statistical test comparing expression level across conditions (e.g., diseased versus normal tissue). It is an alternative to the method described by Ramm and Nadon in “Process for Evaluating Chemical and Biological Assays”, International Application No. PCT/IB99/00734. As such, it can be used in addition to (or in place of) the procedures described by Ramm and Nadon (ibid).
Disadvantages of all prior art procedures include:                1. The value chosen as a normalization reference (e.g., 75th percentile, etc.) is arbitrary;        2. Given that the choice of normalization reference is arbitrary, dividing by the reference value overcorrects some elements and undercorrects others;        3. Because prior art procedures do not estimate systematic error within the context of a statistical model, data transformations that are necessary for correct inferences may not be performed or may be applied incorrectly;        4. Because prior art procedures do not estimate systematic error within the context of a statistical model, normalization can alter the true structure of the data;        5. Because prior art procedures do not detect outliers within the context of a statistical model, true outliers may go undetected and non-outliers may be incorrectly classified as outliers;        6. Classification of values as outliers or not is arbitrary and subjective;        7. Theoretical assumptions about data structure (e.g., that residuals are normally distributed) are not examined empirically.        8. Normalization procedures may create additional measurement error that is not present in the original non-normalized measurements        
The term “treatment condition” will refer to an effect of interest. Such an effect may pre-exist (e.g., differences across different tissues or across time) or may be induced by an experimental manipulation.
Hybridization arrays produced under different treatment conditions may be statistically dependent or independent. Microarray technology in which two different target treatment samples are labelled with different fluors and are then cohybridized onto each arrayed element represent one example of statistical dependence. Typically, expression ratios of the raw signals generated by the two fluors are examined for evidence of differences across treatment conditions.
Chen, Dougherty, & Bittner “Ratio-based decisions and the quantitative analysis of cDNA microarray images”, Journal of Biomedical Optics, 2, 364-374 (1997) have presented an analytical mathematical approach that estimates the distribution of non-replicated differential ratios under the null hypothesis. This approach is similar to the present invention in that it derives a method for obtaining confidence intervals and probability estimates for differences in probe intensities across different conditions. It differs from the present invention in how it obtains these estimates. Unlike the present invention, the Chen et al. approach does not obtain measurement error estimates from replicate probe values. Instead, the measurement error associated with ratios of probe intensities between conditions is obtained via mathematical derivation of the null hypothesis distribution of ratios. That is, Chen et al. derive what the distribution of ratios would be if none of the probes showed differences in measured values across conditions that were greater than would be expected by “chance.” Based on this derivation, they establish thresholds for statistically reliable ratios of probe intensities across two conditions. The method, as derived, is applicable to assessing differences across two conditions only. Moreover, it assumes that the measurement error associated with probe intensities is normally distributed. The method, as derived, cannot accommodate other measurement error models (e.g., lognormal). It also assumes that all measured values are unbiased and reliable estimates of the “true” probe intensity. That is, it is assumed that none of the probe intensities are “outlier” values that should be excluded from analysis. Indeed, outlier detection is not possible with the approach described by Chen et al.
The present invention applies the processes described by Ramm and Nadon in “Process for Evaluating Chemical and Biological Assays”. International Application No. PCT/IB99/00734 and by Ramm, Nadon and Shi in “Process for Removing Systematic Error and Outlier Data and for Estimating Random Error in Chemical and Biological Assays”. Provisional Application No. 60/139,639 (1999) to two or more statistically dependent genomic samples.
The present invention differs from prior art in that:                1. It can accommodate various measurement error models (e.g., lognormal);        2. It can detect outliers within the context of a statistical model;        3. It can be used to examine theoretical assumptions about data structure (e.g., that residuals are normally distributed).        